By S. Fenyo, Thomas Frey

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**Example text**

W e shall develop a determinant of this type in terms o f the cofactors of this column. A l l these cofactors will be (k + l)-order determinants o f the square minors of R. These will all be equal to zero as the order of R is fLk. This means that the determinants in this category also disappear. Thirdly there will be those in which at least two columns contain elements o f a k +1 b*+ j . These determinants will certainly be zero as the columns in question are o f the form a bp,k+l 2,k+\bPtk+i l , k + l aj a n \,k+l ( a bqtk+l a 2,k+lbqk+l which means that they are proportional to each other.

13 THE DYADIC DECOMPOSITION OF CERTAIN IMPORTANT MATRICES 59 If on the other hand b^a we can decompose the product A Β in a similar way as follows: + u2 (v* w>i) H h ua(ν* w,)] z\ + AB = [u1 (v* + · · · + [«i = ki ζ ι + k2z2 + u2(vîWb) + -'+ ua(v*wby] 4- zi = ζ& , h where * / = «l + M e( l ? , 6). W e see immediately from this dyadic decomposition that ρ (A B) ^ b = min (a, b) = min { ρ ( 4 ) , ρ (Β)}. 13 The dyadic decomposition of certain important matrices. a) The decomposition of a projection matrix.

Alkl ^ 1 ···<*»*, —<*nmj u hki \Jtnkt J r a h 1 ··· a ki · · · ii m J W e see immediately that the i^-th row and A^-th column of Ax consist entirely of zeros. If A ί has no non-zero elements and thus A l = 0, we have A1=0 = A 1 whence A = In other words, Λ is itself a dyadic product. (1)* Λ ^ Λ λ* ^ 2 „(l)fc(l)* il k2 where α$ and éj^* are of similar significance to aki, bfx. It then follows that the / 2-th row and k2-th column comprise only zeros. 08 ^a a lk1 iik2 ik 2 «2Ä 2 2 a ~ ^ a ki ii k2 4 iiki 1 Λ) = a a hkl ilk2 a hk2 = o a hkl Λ we see that even the / r t h row of A2 has no non-zero elements.